Refractive boundary elements, devices, and materials

ABSTRACT

An optical device includes an interface between two or more media. The refractive indices, orientations of media, and alignment relative to a propagating wave define a refractive boundary at which reflections may be reduced or eliminated, and at which, for certain incident angles, rays may be refracted on the same side of the normal as the incident ray.

CROSS-REFERENCE TO RELATED APPLICATIONS

The present application is related to, claims the earliest availableeffective filing date(s) from (e.g., claims earliest available prioritydates for other than provisional patent applications; claims benefitsunder 35 USC § 119(e) for provisional patent applications), andincorporates by reference in its entirety all subject matter of thefollowing listed application(s) (the “Related Applications”); thepresent application also claims the earliest available effective filingdate(s) from, and also incorporates by reference in its entirety allsubject matter of any and all parent, grandparent, great-grandparent,etc. applications of the Related Application(s) The United States PatentOffice (USPTO) has published a notice to the effect that the USPTO'scomputer programs require that patent applicants reference both a serialnumber and indicate whether an application is a continuation orcontinuation in part. The present applicant entity has provided below aspecific reference to the application(s) from which priority is beingclaimed as recited by statute Applicant entity understands that thestatute is unambiguous in its specific reference language and does notrequire either a serial number or any characterization such as“continuation” or “continuation-in-part.” Notwithstanding the foregoing,applicant entity understands that the USPTO's computer programs havecertain data entry requirements, and hence applicant entity isdesignating the present application as a continuation in part of itsparent applications, but expressly points out that such designations arenot to be construed in any way as any type of commentary and/oradmission as to whether or not the present application contains any newmatter in addition to the matter of its parent application(s).

RELATED APPLICATIONS

1. United States patent application entitled REFRACTIVE BOUNDARYELEMENTS, DEVICES, AND MATERIALS, U.S. Ser. No. 10/802,100, namingRoderick A. Hyde as inventor, filed 16 Mar., 2004.

TECHNICAL FIELD

The present invention relates to elements, methods, or materials forrefraction.

BACKGROUND

One common type of refraction is the bending of the path of a lightwaveas it crosses a boundary between two media. In conventional optics,Snell's law gives the relationship between the angles of incidence andrefraction for wave crossing the boundary:

n ₁ sin(Θ₁)=n ₂ sin(Θ₂).

This relationship is shown in FIG. 1, where a ray 100 in a first medium101 arrives at a boundary 102 at an angle Θ₁, as referenced to thenormal 104. As the ray 100 crosses the boundary 102, the ray 100 is bentso that it continues propagating as a refracted ray 106 at an angle Θ₂.

While this relatively simple ray optics presentation of refraction iswidely accepted, a more thorough examination of refraction involvesconsideration of propagation of electromagnetic waves and considerationsof energy reflected at a boundary. FIG. 2A represents thisdiagrammatically with an incident wave 108 crossing a boundary 110. Aportion of the energy is reflected as represented by the wave 112 and aportion of the energy propagates as a transmitted wave 114. Asrepresented by the spacing between the waves, the wavelength of thetransmitted wave 114 is shorter than that of the incident wave 108,indicating that the refractive index n₂ experienced by the transmittedwave 114 is higher than the refractive index n₁ experienced by theincident wave 108.

As indicated by the figures and by Snell's law, a lightwave travelingfrom a lower index of refraction to a higher index of refraction will bebent toward the normal at an angle determined by the relative indices ofrefraction. For this system, in a conventional analysis, the range ofrefracted angles relative to the normal is typically confined to a rangefrom 0 degrees to a maximum angle determined by an angle of totalreflection. Additionally, the amount of light energy reflected at theboundary is a function of the relative indices of refraction of the twomaterials.

More recently, it has been shown that under certain limited conditions,rays traveling across a boundary may be refracted on the same side ofthe normal as the incident ray in a phenomenon called “negativerefraction.” Some background on the developments can be found in Pendry,“Negative Refraction Makes a Perfect Lens,” Physical Review Letters,Number 18, Oct. 30, 2000, 3966-3969; Shelby, Smith, and Schultz,“Experimental Verification of a Negative Index of Refraction,” Science,Volume 292, Apr. 6, 2001, 77-79; Houck, Brock, and Chuang, “ExperimentalObservations of a Left-Handed Material That Obeys Snell's Law,” PhysicalReview Letters, Number 13, Apr. 4, 2003, 137401-(1-4); each of which isincorporated herein by reference. With particular reference to negativerefraction, Zhang, Fluegel and Mascarenhas have demonstrated this effectat a boundary between two pieces of YVO₄ crystal, where the pieces ofcrystal are rotated such that the ordinary axis of the first piece isparallel to the extraordinary axis of the second piece. Thisdemonstration was presented in Zhang, Fluegel and Mascarenhas, “TotalNegative Refraction in Real Crystals for Ballistic Electrons and Light,”Physical Review Letters, Number 15, Oct. 10, 2003, 157404-(1-4), whichis incorporated herein by reference. FIG. 2B shows the interface, therelative axes and the nomenclature used in the descriptions herein forthe case of positive refraction. FIG. 2C shows the same aspects fornegative refraction.

The YVO₄ crystal treated by Zhang, et al., is an example of ananisotropic crystal whose dielectric permittivity is defined by thematrix,

$ {\quad\begin{pmatrix}ɛ_{o} & 0 & 0 \\0 & ɛ_{e} & 0 \\0 & 0 & ɛ_{z}\end{pmatrix}}$

and where ∈_(o), ∈_(e), and ∈_({circumflex over (z)}) are not all thesame. In general, the refractive index n of a medium is related to thedielectric permittivity ∈ as,

n=√{square root over (μ∈)},

where μ is the magnetic permeability of the medium.

A more general case, described by Zheng Liu, et al., NEGATIVE REFRACTIONAND OMNIDIRECTIONAL TOTAL TRANSMISSION AT A PLANAR INTERFACE ASSOCIATEDWITH A UNIAXIAL MEDIUM, Phys. Review B (115402), dated Mar. 4, 2004,bearing submission date Oct. 13, 2003, relates to an interface between auniaxial medium and a second medium and is incorporated herein byreference. Liu describes the propagation of waves through uniaxial andisotropic materials to demonstrate reflectionless refraction at aninterface between two uniaxial materials or between a uniaxial materialand an isotropic material.

SUMMARY

In an optical element having materials of differing properties, thematerial properties may define an interface or other transition that mayrefract light traveling through the materials. In one aspect, theelement may include two or more materials that define one or moreboundaries. The material properties and/or orientation may be selectedto establish refraction at the boundary. In one aspect, the materialproperties are refractive indices or dielectric constants. Theproperties may be selected so that refraction at the boundary issubstantially reflectionless.

In one approach, materials are selected to define a boundary. Thematerials are selected with refractive indices that are relatedaccording to the geometric means of their constituents. In one approach,the constituents are ordinary and extraordinary indices of refraction.In one approach, at least one of the materials is an anisotropicmaterial. In one approach two or more of the materials are anisotropic.In one approach one or more of the materials is anisotropic according toits index of refraction. Where one or more of the materials isanisotropic, the anisotropic materials are oriented such that thefollowing boundary conditions are satisfied by a wave incident on theboundary:

{circumflex over (z)}·{right arrow over (H)} ₁ ={circumflex over(z)}·{right arrow over (H)} ₂

ŷ·{right arrow over (E)} ₁ =ŷ·{right arrow over (E)} ₂

ŷ·{right arrow over (k)} ₁ =ŷ·{right arrow over (k)} ₂

In one approach, the materials have dielectric constants andorientations that satisfy the relationship:

β₁ɛ₁² = β₂ɛ₂²${{\sqrt{\beta_{1}}\cos^{2}\varphi_{1}} + {\frac{1}{\sqrt{\beta_{1}}}\sin^{2}\varphi_{1}}} = {{\sqrt{\beta_{2}}\cos^{2}\varphi_{2}} + {\frac{1}{\sqrt{\beta_{2}}}\sin^{2}\varphi_{2}}}$

In one case, one or more of the materials is a biaxial material. Inaddition to the foregoing, various other method and/or system aspectsare set forth and described in the text (e.g., claims and/or detaileddescription) and/or drawings of the present application.

The foregoing is a summary and thus contains, by necessity;simplifications, generalizations and omissions of detail; consequently,those skilled in the art will appreciate that the summary isillustrative only and is NOT intended to be in any way limiting. Otheraspects, inventive features, and advantages of the devices and/orprocesses described herein, as defined solely by the claims, will becomeapparent in the non-limiting detailed description set forth herein.

BRIEF DESCRIPTION OF THE FIGURES

FIG. 1 is a ray diagram showing propagation of a light ray across aboundary between two media.

FIG. 2A is a propagation diagram showing propagation and reflection of awave at a boundary.

FIG. 2B is a diagram showing a boundary and nomenclature associated withbeam propagation.

FIG. 2C is a diagram showing a boundary and nomenclature associated withbeam propagation for negative refraction.

FIG. 3 is a representation of a wave propagating across the boundarybetween two different media having equivalent geometric means of theirordinary and extraordinary indices of refraction.

FIG. 4 is a representation of a three layer structure with a raypropagating through two interfaces between the layers.

FIG. 5 is a representation of a structure having five layers, eachhaving a respective index of refraction.

FIG. 6 is a side view of a stack of three materials having non-uniformthicknesses.

FIG. 7 is a side view of a stack of three materials, where the centralmaterial has a graded index of refraction.

FIG. 8 is a diagrammatic representation of a pair of materials, whereone of the materials is a dynamically controllable material.

FIG. 9 is a diagrammatic representation of a stack of materials wherethe central material is a dynamically controllable material.

FIG. 10 is a diagrammatic representation of a circulator and waveguide.

FIG. 11 is a representation of a manmade material having selectedelectromagnetic properties.

DETAILED DESCRIPTION

As shown in FIG. 3, a first ray 300 travels through a first medium 302with a first index of refraction n₁ at an angle Θ₁ relative to a normal306. The index of refraction n₁ that ray 300 sees may depend on theordinary and extraordinary indices of refraction of the medium, and alsoupon the angle the ray 300 makes with the ordinary axis ô andextraordinary axis ê if the medium is anisotropic in the {circumflexover (x)}, ŷ plane, as defined below. The ray 300 arrives at a boundary304 between the first medium 302 and a second medium 308. As the ray 300crosses the boundary 304, it is refracted at an angle Θ₂ that is afunction of the index of refraction n₂ of the second medium 308, whereagain, n₂ depends on the ordinary and extraordinary indices ofrefraction of the medium, and also depends on the angle the ray 300makes with the ordinary and extraordinary axes ô and ê if the medium isanisotropic in the {circumflex over (x)}, ŷ plane. While the angle ofrefraction is shown as negative, the discussion herein may be relatedequally to positive refraction in most cases.

The following discussion assumes one of the principal axes of the firstmedium 302 (defined to be {circumflex over (z)}₁) is aligned with one ofthe principal axes of the second medium 308 (defined to be {circumflexover (z)}₂). A further condition for purposes of this exemplarydiscussion is that this common direction (defined as {circumflex over(z)}₁={circumflex over (z)}₂={circumflex over (z)}) lies in the plane ofthe interface between the two media 302, 308. A Cartesian basis set({circumflex over (x)}, ŷ, {circumflex over (z)}) can be defined suchthat {circumflex over (x)} is perpendicular to the plane of the boundary304, and the vectors y and {circumflex over (z)} are parallel to it. Theordinary axis ô and extraordinary axis ê experienced by the ray 300 arealso in the {circumflex over (x)}, ŷ plane. (It is important to notethat, although ô and ê are used to represent an ordinary andextraordinary axis, this is not meant to imply a uniaxial material; thedielectric constant in the {circumflex over (z)} direction can be almostanything depending upon the conditions to be satisfied.) For thestructure described for this exemplary embodiment, light rays thatexperience positive refraction, negative refraction and/orreflectionless refraction travel in the {circumflex over (x)}, ŷ planeand are also polarized so that the electric displacement {right arrowover (D)} lies in the {circumflex over (x)}, ŷ plane and the magneticfield {right arrow over (H)} lies in the {circumflex over (z)}direction. This configuration is structured to allow simplifieddiscussion of refraction confined to the {circumflex over (x)}, ŷ planeand should not be considered to be limiting.

Note that the indices of refraction n₁, n₂ may be simplifications of theactual index of refraction, because the index of refraction of eachmedium may depend upon the direction of the wave traveling through themedium. For example, if the first medium 302 is an anisotropic medium,the index of refraction n₁ experienced by a wave traveling through themedium may depend upon an ordinary component n_(1o) and an extraordinarycomponent n_(1e). Similarly, if the second medium is an anisotropicmedium, it may also have ordinary and extraordinary indices n_(2o),n_(2e). On the other hand, if the medium is isotropic, the ordinary andextraordinary indices of refraction will be the same.

In a simplified case where the first and second media 302, 308 are thesame isotropic material, the index of refraction experienced by apropagating wave will remain constant across the boundary and the wavewill propagate un-refracted. Additionally, because the refractive indexdoes not change, no energy will be reflected at the boundary.

In a slightly more complex case, similar to that of the Zhang paper, thefirst and second media may be the same anisotropic material. It has beenshown by Zhang et al., that if certain conditions are met, the amount ofenergy reflected at the boundary will be zero for identical anisotropicmaterials when the materials are rotated relatively such that theordinary axis of the first media 302 is inclined at the negative of theangle defined by the ordinary axis of the second media 308. However,because a given polarization component of the wave experiences a changein index of refraction, the lightwave will be refracted.

In the more general case similar to that described in Zheng, the media302, 308 on either side of the boundary 304 of FIG. 3 are not limited toidentical, anisotropic media. In one case, the first material 302 isisotropic in the {circumflex over (x)}, ŷ plane, having equal ordinaryand extraordinary indices of refraction n_(1o), n_(1e). The secondmedium 308 is anisotropic in the {circumflex over (x)}, ŷ plane, havingan ordinary index of refraction n_(2o) different from its extraordinaryindex of refraction n_(2e).

For the case where the first medium 302 is isotropic in the {circumflexover (x)}, ŷ plane and the second medium 308 is anisotropic in the{circumflex over (x)}, ŷ plane, the indices of refraction of the twomedia 302, 308 are related by their geometric mean according to therelationship:

n ₁=√{square root over (n _(2o) ·n _(2e))}

In this embodiment, the magnetic permeability μ of both media is a) thesame in both media, and b) isotropic in both media. Defining thepermittivities by,

∈_(1o)≡∈₁, ∈_(e1)≡β₁∈₁, ∈_(2o)≡∈₂, ∈₂₃≡β₂∈₂

then the permittivities (for the case where the first medium 302 isisotropic in the x, y plane and the second medium 308 is anisotropic inthe {circumflex over (x)}, ŷ plane) are related as,

∈₁=√{square root over (β₂∈₂ ²)}

This relationship can be derived by satisfying certain boundaryconditions and a dispersion relation for zero reflected energy.

The fields in each material must satisfy Maxwell's relations. These leadto standard expressions for the fields and a dispersion relation for thewave vector:

$\overset{\rightarrow}{H} = {H\hat{z}}$$\overset{\rightarrow}{E} = {\frac{kH}{\omega \; ɛ}\left\{ {{\hat{e}\frac{\sin \; \phi}{\beta}} - {\hat{o}\cos \; \phi}} \right\}}$$k^{2} = {\mu \; ɛ\; {\omega^{2} \cdot \frac{\beta}{{\beta \; \cos^{2}\phi} + {\sin^{2}\phi}}}}$

Where, for an anisotropic material, the electric field {right arrow over(E)} is related to the electric displacement {right arrow over (D)} andthe dielectric tensor ∈_(x,y) by:

$\begin{pmatrix}D_{x} \\D_{y} \\D_{z}\end{pmatrix} = {\begin{pmatrix}ɛ_{xx} & ɛ_{xy} & ɛ_{xz} \\ɛ_{yx} & ɛ_{yy} & ɛ_{yz} \\ɛ_{zx} & ɛ_{zy} & ɛ_{zz}\end{pmatrix}{\begin{pmatrix}E_{x} \\E_{y} \\E_{z}\end{pmatrix}.}}$

As represented in FIG. 2B, k represents the magnitude of the wave numberand φ represents the angle between {right arrow over (k)} and theprincipal axis of the respective material 302 or 308. The Poyntingvector {right arrow over (P)} is defined in the usual way by,

{circumflex over (P)}={right arrow over (E)}×{right arrow over (H)}.

For reflectionless refraction to occur, a wave incident on the boundaryis completely transmitted from the first medium 302 to the second medium308, and no portion of the wave is reflected back into the first medium302. This implies a set of continuity equations at the boundary. Amongthese are:

{circumflex over (z)}·{right arrow over (H)} ₁ ={circumflex over(z)}·{right arrow over (H)} ₂

ŷ·{right arrow over (E)} ₁ =ŷ·{right arrow over (E)} ₂

ŷ·{right arrow over (k)} ₁ =ŷ·{right arrow over (k)} ₂

For the geometry in FIG. 2B, these can be written as:

$\begin{matrix}{{\frac{1}{\beta_{2}ɛ_{2}}\left\{ {{\beta_{2}\cos \; \varphi_{2}d_{2}} + {\sin \; \varphi_{2}h_{2}}} \right\}} = {\frac{1}{\beta_{1}ɛ_{1}}\left\{ {{\beta_{1}\cos \; \varphi_{1}d_{1}} + {\sin \; \varphi_{1}h_{1}}} \right\}}} \\{{{\sin \; \varphi_{2}d_{2}} - {\cos \; \varphi_{2}h_{2}}}} \\{= {{\sin \; \varphi_{1}d_{1}} - {\cos \; \varphi_{1}h_{1}}}}\end{matrix}$

using:

d ₁ =k ₁ cos φ₁ , h ₁ ≡k ₁ sin φ₁, d₂ ≡k ₂ cos φ₂ , h ₂ ≡k ₂ sin φ₂

Satisfying the dispersion relation in each medium produces:

${\frac{1}{\beta_{2}ɛ_{2}}\left( {{\beta_{2}d_{2}^{2}} + h_{2}^{2}} \right)} = {\frac{1}{\beta_{1}ɛ_{1}}{\left( {{\beta_{1}d_{1}^{2}} + h_{1}^{2}} \right).}}$

The above equations are satisfied for arbitrary incident angles (in the{circumflex over (x)}, ŷ plane) under the following conditions:

β₁ɛ₁² = β₂ɛ₂²${{\sqrt{\beta_{1}}\cos^{2}\varphi_{1}} + {\frac{1}{\sqrt{\beta_{1}}}\sin^{2}\varphi_{1}}} = {{\sqrt{\beta_{2}}\cos^{2}\varphi_{2}} + {\frac{1}{\sqrt{\beta_{2}}}\sin^{2}\varphi_{2}}}$

If the first medium 302 is isotropic, then β₁=1, and the condition thatβ₁∈₁ ²=β₂∈₂ ² reduces to ∈₁=√{square root over (β₂∈₂ ²)}.

While the embodiment described above relates to a ray traveling from anisotropic medium to an anisotropic medium, propagation in the oppositedirection (i.e., from an anisotropic medium to an isotropic medium) mayalso be within the scope of the invention.

In another embodiment, both the first medium 302 and the second medium308 are anisotropic in the {circumflex over (x)}, ŷ plane, though theyare different media. However, the two media are selected such that thegeometric means of their ordinary and extraordinary indices ofrefraction are substantially equal, as can be represented by:

n_(1o)n_(1e)=n_(2o)n_(2e)

or, symmetrically:

β₁∈₁ ²=β₂∈₂ ².

The analysis for this circumstance is the more general portion of theanalysis above, though not simplified by assuming the first medium 302is isotropic. Note that this approach is not necessarily limited tocases where the permittivity is symmetrical. That is, one or both of thematerials need not be uniaxial or isotropic. Instead, the approach maybe generalized to cover materials where the permittivity tensors are notlimited to those corresponding to uniaxial or isotropic materials. Inone particularly interesting case one or both of the materials mayinclude biaxial materials. The biaxial materials may be naturallyoccurring or manmade materials.

One skilled in the art will recognize that many naturally occurringmedia of different material types will not have equal geometric means oftheir ordinary and extraordinary indices of refraction. In part becausethe above relationships do not necessarily require that the two media beof the same material, this aspect may be reduced somewhat. For example,the index of refraction of some man-made materials can be controlled tosome extent. This ability can be used to more closely satisfy therelationships described above. Moreover, in some materials, the ordinaryand/or index of refraction may depend upon the exterior circumstances,such as applied magnetic or electrical fields.

Further, one skilled in the art will recognize that extremely precisecontrol of the index of refraction may be difficult, and that many ofthe approaches or benefits described herein may be realized bysubstantially complying with the geometric mean and other relationshipsof the above equations, rather than exactly satisfying therelationships.

The above approach is not limited to a single boundary between twomaterials. For example, as shown in FIG. 4, a lightwave 400 travelingthrough a first medium 402 crosses a boundary 404 and enters a secondmedium 406. The lightwave 400 produces a refracted wave 408 thatpropagates through the second medium 406 to a boundary 410. At theboundary 410, the refracted wave 408 enters a third medium 412 toproduce a second refracted wave 414. At each of the boundaries 404, 410,the amount of energy reflected depends upon the relative indices ofrefraction on opposite sides of the boundary. At the first boundary 404,the geometric mean of the ordinary and extraordinary indices ofrefraction n_(1o), n_(1e) in the first medium 402 equals the geometricmean of the ordinary and extraordinary indices of refraction n_(2o),n_(2e) in the second medium 406.

Similarly, at the second boundary 410, the geometric mean of theordinary and extraordinary indices of refraction n_(2o), n_(2e) in thesecond medium 406 equals the geometric mean of the ordinary andextraordinary indices of refraction n_(3o), n_(3e) of the third medium412.

The above approach may be extended to a larger number of layers, asrepresented by FIG. 5. Once again, the geometric means of the indices ofrefraction on opposite sides of boundaries 500, 502, 504, 506 aresubstantially equal. While the layers in FIG. 5 are represented asrectangular and having uniform thickness, the approach here is notnecessarily so limited. For example, the thickness of the layers mayvary from layer to layer. Further, individual layers may havenon-uniform thicknesses, as represented by the trio of wedges 602, 604,606 shown in FIG. 6.

In still another approach, presented in FIG. 7, a first section ofmaterial 702 has an ordinary index of refraction n_(1o) and anextraordinary index of refraction n_(1e). A second section of material704 abuts the first material 702 and defines a first interface 708. Thesecond section of material 704 has a gradient index of refraction thatbegins at a left index of refraction n_(2L) and changes to a right indexof refraction n_(2R). This discussion assumes that the second materialis isotropic for clarity of presentation, however, the embodiment is notnecessarily so limited. In some applications, it may be desirable forthe second material 704 to be a gradient index, anisotropic material.This would likely be achieved with manmade materials.

A third material 706 abuts the second material 704 and defines a secondinterface 710. The third material 706 has an ordinary index ofrefraction n_(3o) and an extraordinary index of refraction n_(3e). Inthis structure, the second material 704 provides a transitional layerbetween the first material 702 and the third material 706. For indexmatching at the first interface, the left index of refraction n_(2L)equals the geometric mean of the first ordinary index of refractionn_(1o) and the first extraordinary index of refraction n_(1e).Similarly, for index matching at the second interface, the right indexof refraction n_(2R) equals the geometric mean of the third ordinaryindex of refraction n_(3o) and the third extraordinary index ofrefraction n_(3e). As above, the approach may be extended to more layersor may be combined with other embodiments to address designconsiderations. For example, a structure may include more than threelayers and may have non-parallel faces. In another aspect, an embodimentmay include a series of layers or discrete elements to form a complexelement. One of skill in the art can adapt known optical designtechniques or computer based programs to design optical elementsimplementing one or more aspects of the herein-described embodiments.Particular cases may include a series of optical elements for enlarging,transmitting, or aligning an image, such as in microscopes, displayviewing optics, or photolithography.

Moreover, one or more of the layers or portions of the layers mayinclude a polarization rotating structure. Such structures are known andmay be passive or active. In active polarization rotating structures,the amount of polarization rotation can be controlled externally, by forexample, application of electric fields. In such approaches, eitheractive or passive, waves propagating through the rotating structure canrotate in polarization in an amount determined by the structure and/orand applied input, thereby providing an additional degree of designfreedom. In one respect, from the point of view of wave polarization,the materials following the rotational structure are effectively rotatedrelative materials preceding the rotating structure.

While the previous embodiments have been described in the context of aconstant set of refractive indices, dynamic adjustment may also bewithin the scope of the invention. For example, as shown in FIG. 8, afirst material 800 abuts a second, dynamically controllable material802. In one embodiment, the dynamically controllable material 802 is anelectrooptic material. As is known, electrooptic materials, such asLiNbO₃, have indices of refraction that depend upon applied electricfields. As represented by a pair of electrodes 804 and 806 and a voltagesource 808, an electric field E may be applied to the second material802. While the electric field E is presented as being applied transverseto a propagating ray 810, other directional applications may beappropriate depending upon the electro optic tensor of the particularmaterial. The dynamically varied index of refraction also need not beconstant. For example, in an electrooptic device, the electric field canbe varied spatially to produce a gradient.

In another embodiment, the dynamically controllable material 802 may bea polarization rotating material, as represented in the structure ofFIG. 9. In this structure, the dynamically controllable material 802provides a transitional layer between a first material 900 and a thirdmaterial 902. The first material 900 has an ordinary index of refractionn_(1o) and an extraordinary index of refraction n_(1e) whose geometricmean matches that of the ordinary and extraordinary indices ofrefraction n_(3o), n_(3e) of the third material 902. However, for propermatching of the materials, it may be desirable that the polarization oflight 904 exiting the first material 900 be rotated before entering thethird material 902. The dynamically controllable material 802 can rotatea polarization of the light 904, responsive to an applied electric fieldE, represented by a voltage source V and respective electrodes 910, 908.For example, dynamic control may be used to establish conditions forsubstantially reflectionless refraction between the dynamicallycontrollable material 802 and the first or second materials 900, 902.

Alternatively, the index of refraction or other material property of thedynamically controllable material 802 may be varied to direct thepropagating energy out of plane. It should be noted that the layers ofmaterials described herein are shown with exemplary aspect ratios andthicknesses. However, the invention is not so limited. The materials maybe made almost arbitrarily thin, on the order of one or a fewwavelengths in many applications, thereby allowing the optical elementsand structures described herein to be very thin.

Although the embodiment described above with respect to polarizationrotation presumes that the first and third materials have the samegeometric mean of their indices of refraction, it may be useful in someapplications to have the geometric means be different. One approach tothis is to combine the transitional index approach of FIG. 7 or theactive control approach of FIG. 8 to rotate polarization and transitionbetween differing geometric mean index of refractions. Similarly, theapproaches of FIGS. 7 and 8 can be combined to allow active control andtransition.

Note that while the dynamically controllable materials of FIGS. 8 and 9respond to electric fields, other dynamically controllable materials maybe within the scope of the invention. For example, some materialsrespond to magnetic fields, temperature, optical energy, stress, or avariety of other inputs. Moreover, the dynamically controllablematerials may be incorporated into other structures including thestructures described earlier with respect to FIGS. 5 and 6.

In addition to linear stacks of materials, other configurations may bewithin the scope of the invention. For example, as representeddiagrammatically in FIG. 10, a series of wedges 1000, 1002, 1004, 1006,1008, 1010, 1012, may be arranged in a geometric structure, such as agenerally circular structure or generally polygonal structure. While thestructure of FIG. 10 includes seven wedges for clarity of presentation,the actual number in arrangement of wedges may be larger or smallerdepending upon the desired result and the amount of refraction at eachinterface.

In the structure of FIG. 10, the refraction at the interfaces isselected such that a ray 1014 propagating through one of the wedgescompletes a relatively circular loop. The polygonal structure 1016 thusforms a resonator ring. Optical resonators are useful in a variety ofapplications, including as resonators for lasers, filters, gyroscopes,or switches as described, for example, in U.S. Pat. No. 6,411,752entitled VERTICALLY COUPLED OPTICAL RESONATOR DEVICES OVER A CROSS-GRIDWAVEGUIDE ARCHITECTURE to Little, et al.; WO0048026A1 entitled OPTICALWAVEGUIDE WAVELENGTH FILTER WITH RING RESONATOR AND 1×N OPTICALWAVEGUIDE WAVELENGTH FILTER to Chu, et al., published Aug. 17, 2000;U.S. Pat. No. 6,052,495 entitled RESONATOR MODULATORS AND WAVELENGTHROUTING SWITCHES to Little, et al.; U.S. Pat. No. 6,603,113 to Numaientitled GYRO COMPRISING A RING LASER IN WHICH BEAMS OF DIFFERENTOSCILLATION FREQUENCIES COEXIST AND PROPAGATE IN MUTUALLY OPPOSITECIRCULATION DIRECTIONS, DRIVING METHOD OF GYRO, AND SIGNAL DETECTINGMETHOD; and in Liu, Shakouri, and Bowers, “Wide Tunable Double RingResonator Tuned Lasers,” IEEE Photonics Technology Letters, Vol. 14, No.5, (May 2002) each of which is incorporated herein by reference.

While the embodiments described above focus primarily on opticalelements and incorporate optical terminology in many cases, thetechniques, structures, and other aspects according to the invention arenot so limited. For example, in some applications, it may be desirableto implement structures configured for other portions of theelectromagnetic spectrum. For example, as shown in FIG. 11, the inputenergy may be radio frequency (RF). In this structure, a first material1100 is a manmade structure including periodically positioned RFstructures 1102. The first material has dielectric constants∈_(1o)-∈_(1e) corresponding respectively to its ordinary andextraordinary axes. The first material 1100 adjoins a second material1104 having dielectric constants ∈_(2o), ∈_(2e) correspondingrespectively to its ordinary and extraordinary axes. Manmade materialshaving anisotropic sets of dielectric constants are known. For example,such materials have been shown to operate at around 10 GHz.

In the embodiment of FIG. 11, the adjoining materials are such that theysatisfy the equation:

β₁ɛ₁² = β₂ɛ₂²${{\sqrt{\beta_{1}}\cos^{2}\varphi_{1}} + {\frac{1}{\sqrt{\beta_{1}}}\sin^{2}\varphi_{1}}} = {{\sqrt{\beta_{2}}\cos^{2}\varphi_{2}} + {\frac{1}{\sqrt{\beta_{2}}}\sin^{2}\varphi_{2}}}$

In one embodiment, each of the materials has a respective anisotropicdielectric constant, such that:

∈₁₀≠∈_(1e)

and

∈₂₀≠∈_(2e)

In another embodiment, the first material is isotropic, such that:

∈_(1o)=∈_(1e)

As described with respect to previous embodiments, the electromagneticwave, though shown propagating in one direction may propagate in theopposite direction and still satisfy the above configuration, for manystructures. While the descriptions above have generally referred tospecific materials or manmade analogs, in some applications one or morelayers of isotropic materials may be air, vacuum, gas, liquid or othersubstances, including substances having a unity index of refraction. Insuch cases, the anisotropic material may have an index less than unity.For example, a material having aligned metal fibers in an isotropicdielectric may have an extraordinary index less than 1. The netgeometric mean of the material in certain orientations can then equalsubstantially 1.

While the exemplary embodiments of FIGS. 1-11 are presented withreference to optical systems and terminology, those skilled in the artwill recognize that at least a portion of the devices and/or processesdescribed herein can apply to other types of systems, including RF,X-ray, or other electromagnetic elements, processes, or systems.

The foregoing detailed description has set forth various embodiments ofthe devices and/or processes via the use of block diagrams, diagrammaticrepresentations, and examples. Insofar as such block diagrams,diagrammatic representations, and examples contain one or more functionsand/or operations, it will be understood as notorious by those withinthe art that each function and/or operation within such block diagrams,diagrammatic representations, or examples can be implemented,individually and/or collectively, by a wide range of hardware,materials, components, or virtually any combination thereof.

Those skilled in the art will recognize that it is common within the artto describe devices and/or processes in the fashion set forth herein,and thereafter use standard engineering practices to integrate suchdescribed devices and/or processes into elements, processes or systems.That is, at least a portion of the devices and/or processes describedherein can be integrated into an optical, RF, X-ray, or otherelectromagnetic elements, processes or systems via a reasonable amountof experimentation.

Those having skill in the art will recognize that a typical opticalsystem generally includes one or more of a system housing or support,and may include a light source, electrical components, alignmentfeatures, one or more interaction devices, such as a touch pad orscreen, control systems including feedback loops and control motors(e.g., feedback for sensing lens position and/or velocity; controlmotors for moving/distorting lenses to give desired focuses). Suchsystems may include image processing systems, image capture systems,photolithographic systems, scanning systems, or other systems employingoptical, RF, X-ray or other focusing or refracting elements orprocesses.

The foregoing described embodiments depict different componentscontained within, or connected with, different other components. It isto be understood that such depicted architectures are merely exemplary,and that in fact many other architectures can be implemented whichachieve the same functionality. In a conceptual sense, any arrangementof components to achieve the same functionality is effectively“associated” such that the desired functionality is achieved. Hence, anytwo components herein combined to achieve a particular functionality canbe seen as “associated with” each other such that the desiredfunctionality is achieved, irrespective of architectures or intermedialcomponents. Likewise, any two components so associated can also beviewed as being “operably connected” or “operably coupled” to each otherto achieve the desired functionality.

While particular embodiments of the present invention have been shownand described, it will be understood by those skilled in the art that,based upon the teachings herein, changes and modifications may be madewithout departing from this invention and its broader aspects and,therefore, the appended claims are to encompass within their scope allsuch changes and modifications as are within the true spirit and scopeof this invention. Furthermore, it is to be understood that theinvention is solely defined by the appended claims. It will beunderstood by those within the art that, in general, terms used herein,and especially in the appended claims (e.g., bodies of the appendedclaims) are generally intended as “open” terms (e.g., the term“including” should be interpreted as “including but not limited to,” theterm “having” should be interpreted as “having at least,” the term“includes” should be interpreted as “includes but is not limited to,”“comprise” and variations thereof, such as, “comprises” and “comprising”are to be construed in an open, inclusive sense, that is as “including,but not limited to,” etc.). It will be further understood by thosewithin the art that if a specific number of an introduced claimrecitation is intended, such an intent will be explicitly recited in theclaim, and in the absence of such recitation no such intent is present.For example, as an aid to understanding, the following appended claimsmay contain usage of the introductory phrases “at least one” and “one ormore” to introduce claim recitations. However, the use of such phrasesshould not be construed to imply that the introduction of a claimrecitation by the indefinite articles “a” or “an” limits any particularclaim containing such introduced claim recitation to inventionscontaining only one such recitation, even when the same claim includesthe introductory phrases “one or more” or “at least one” and indefinitearticles such as “a” or “an” (e.g., “a” and/or “an” should typically beinterpreted to mean “at least one” or “one or more”); the same holdstrue for the use of definite articles used to introduce claimrecitations. In addition, even if a specific number of an introducedclaim recitation is explicitly recited, those skilled in the art willrecognize that such recitation should typically be interpreted to meanat least the recited number (e.g., the bare recitation of “tworecitations,” without other modifiers, typically means at least tworecitations, or two or more recitations).

1-38. (canceled)
 39. A method of producing a refractive elementincluding one or more biaxial materials, comprising: identifying a firstelectromagnetic parameter of a first material corresponding to aselected first internal axis of the first material; identifying a secondelectromagnetic parameter of the first material corresponding to aselected second internal axis of the first material, the secondelectromagnetic parameter being related to the first electromagneticparameter by a ratio β₁; identifying a first electromagnetic parameterof a second material corresponding to a selected first internal axis ofthe second material, the second material being different material fromthe first material; identifying a second electromagnetic parameter ofthe second material corresponding to a selected second internal axis ofthe second material, the second electromagnetic parameter of the secondmaterial being related to the first electromagnetic parameter of thesecond material by a ratio β₂, such that the ratio of the firstelectromagnetic parameter of the second material to the firstelectromagnetic parameter of the first material is equal to,$\sqrt{\frac{\beta_{1}}{\beta_{2}}};$ determining an orientation of thesecond material relative to the first material at which reflected energyat a boundary between the first and second materials is substantiallyzero; orienting the materials at the determined orientation; and joiningthe materials at the determined orientation.
 40. The method of claim 39wherein the first electromagnetic parameter of the first material is apermittivity ∈₁ of the first material and the first electromagneticparameter of the second material is a permittivity ∈₂ of the secondmaterial.
 41. The method of claim 40 wherein the selected first internalaxis of the first material is a principal axis of the material.
 42. Themethod of claim 41 wherein the selected first internal axis of thesecond material is a principal axis of the material
 43. The method ofclaim 42 wherein determining an orientation of the second materialrelative to the first material an angle at which reflected energy at aboundary between the first and second materials is substantially zeroincludes orienting the first material principal axis and second materialprincipal axis according to the equation:${{\sqrt{\beta_{1}}\cos^{2}\varphi_{1}} + {\frac{1}{\sqrt{\beta_{1}}}\sin^{2}\varphi_{1}}} = {{\sqrt{\beta_{2}}\cos^{2}\varphi_{2}} + {\frac{1}{\sqrt{\beta_{2}}}\sin^{2}\varphi_{2}}}$44-69. (canceled)
 70. A refractive element produced by the method ofclaim
 39. 71. A refractive element produced by the method of claim 44.72. A method, comprising: interposing a refractive element in a path ofan electromagnetic wave, wherein the refractive element has twodifferent materials disposed on opposite sides of an interface, whereineach material has a pair of optical axes and a respective pair ofdielectric constants associated with the pair of optical axes, whereinat least one of the materials is a biaxial material, and wherein thepairs of optical axes of the two different materials are rotatedrelative to each other so that a magnetic field component andcorresponding orthogonal electric field and wave vector components ofthe electromagnetic wave in the plane of the interface remainsubstantially unchanged as the electromagnetic wave passes through theinterface.
 73. The method of claim 72 wherein interposing the refractiveelement comprises interposing a refractive element in which a product ofthe pair of dielectric constants of the biaxial material issubstantially the same as a product of the pair of dielectric constantsof the material on the opposite side of the interface.
 74. A method ofproducing a refractive element, comprising: joining a first material anda second material to form a common interface, wherein each material hasa pair of optical axes and a respective pair of dielectric constantsassociated with the pair of optical axes, wherein at least one of thematerials is a biaxial material, and wherein the pairs of optical axesof the two joined materials are rotated relative to each other so thatwhen an electromagnetic wave is incident on the common interface—amagnetic field component of the electromagnetic wave in a plane of theinterface and corresponding orthogonal electric field and wave vectorcomponents of the electromagnetic wave in the plane of the interfaceremain substantially unchanged as the electromagnetic wave passesthrough the interface.
 75. The method of claim 74 wherein a product ofthe pair of dielectric constants of the first material is substantiallythe same as a product of the pair of dielectric constants of the secondmaterial.
 76. A refractive element produced by the method of claim 74.77. A refractive element produced by the method of claim 75.